An evangelical atheist is simply any atheist who actively tries to convert religious people to atheism.
It is important to note that, contrary to popular belief, atheists almost never make a definitive claim that "god does not exist". Only a very small percentage of atheists make such claims and they are referred to as "strong atheists".
"Belief that god does not exist" is not the correct definition of atheism, even though many dictionaries incorrectly list it as such. More accurately, atheism actually means something more like "the belief that there is insufficient evidence to assume the existence of god (over any alternatives), and thus one should not live as though one knows that god exists". Thus it is never fair to say that atheists have "just as much faith" as religious people do.
No human being has reliable evidence of Santa Claus existing, but that does not mean that one can definitively say that he does not exist, because it would essentially require searching the entire universe for him, which is impossible. There is no good evidence for Santa Claus, thus we shouldn't live our lives in such a way that we assume he is actually real. Atheists think about god in the same way.
Some atheists are evangelical is because they believe that the world would be better off if people didn't make assumptions that are not justified. Modern science came into existence only because a certain portion of human beings decided that conclusions about the world must be well justified.
If an atheist tries to actively convert you to atheism, then that atheist could be called an evangelical atheist.